Basic Definitions and The Spectral Estimation Problem

Transcription

1 Informal Definition of Spectral Estimation Given: A finite record of a signal. Basic Definitions and The Spectral Estimation Problem Determine: The distribution of signal power over frequency. signal t=,,... t spectral density ω ω+δω π ω Lecture! = (angular) frequency in radians/(sampling interval) f =!= = frequency in cycles/(sampling interval) Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

2 t=; jy(t)j < Applications Deterministic Signals Temporal Spectral Analysis Vibration monitoring and fault detection Hidden periodicity finding Speech processing and audio devices fy(t)g t=; discretetime deterministic data = sequence Medical diagnosis If: X Control systems design Then: Y (!) = X t=; y(t)e ;i!t Seismology and ground movement study Radar, Sonar exists and is called the DiscreteTime Fourier Transform (DTFT) Spatial Spectral Analysis Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L 4 Source location using sensor arrays

3 ; S(!)d! Energy Spectral Density Random Signals Parseval's Equality: X Z Random Signal random signal probabilistic statements about future variations jy(t)j = t=; t where current observation time S(!) 4 = jy (!)j = Energy Spectral Density Here: X t=; jy(t)j = We can write But: E n jy(t)j o < S(!) = X k=; (k)e ;i!k E fg = Expectation over the ensemble of realizations where E n jy(t)j o = Average power in y(t) (k) = X t=; y(t)y (t ; k) PSD = (Average) power spectral density Lecture notes to accompany Introduction to Spectral Analysis Slide L 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L

4 = r(k) = E n jy(t)j o = r(0)! d! NX ;i!t 9 >= y(t)e > First Definition of PSD Second Definition of PSD k=; r(k)e ;i!k (!) = X where r(k) is the autocovariance sequence (ACS) (!) = 8 >< r(k) = E fy(t)y (t ; k)g lim E N! r(k) = r (;k) Note that Interpretation: Z r(0) jr(k)j ; (!)ei!k d! (Inverse DTFT) Note that where (!) = lim E N jy N(!)j N! NX >: N t= Z Y N (!) = t= y(t)e ;i!t so ; (!)d! is the finite DTFT of fy(t)g. infinitesimal signal power in the band (!)d! = Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L 8

5 e(t) e (!) y(t) (!) jh(!)j H(q) e (!) = y X X Properties of the PSD Transfer of PSD Through Linear Systems System Function: H(q) = X k=0 h k q ;k P: (!) = (! + ) for all!. Thus, we can restrict attention to where q ; = unit delay operator: q ; y(t) = y(t ; )! [; ] () f [;= =] P: (!) 0 Then P: If y(t) is real, y(t) = Then: (!) = (;!) k=0 h k e(t ; k) Otherwise: H(!) = k=0 h k e ;i!k (!) = (;!) y (!) = jh(!)j e (!) Lecture notes to accompany Introduction to Spectral Analysis Slide L 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L 0

6 The Spectral Estimation Problem The Problem: From a sample Find an estimate of (!): Two Main Approaches : f^(!)! [; ]g Nonparametric: Derived from the PSD definitions. Periodogram and Correlogram Methods fy() ::: y(n)g Parametric: Lecture Assumes a parameterized functional form of the PSD Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

7 ^ p (!) = N 8 >< >: NX NX ;i!t 9 >= y(t)e > Periodogram Correlogram Recall nd definition of (!): (!) = lim E N! Given : fy(t)g N t= Recall st definition of (!): t= N Drop lim N! and E fg toget (!) = X k=; r(k)e ;i!k Truncate the P and replace r(k) by ^r(k) : y(t)e ;i!t ^ c (!) = X N; t= k=;(n;) ^r(k)e ;i!k Natural estimator Used by Schuster (900) to determine hidden periodicities (hence the name). Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

8 = ^r(k) ; k N NX NX ^ p (!) = N NX ^ p (!) = ^ c (!) Covariance Estimators (or Sample Covariances) Relationship Between ^ p (!) and ^ c (!) If: the biased ACS estimator ^r(k) is used in ^ c (!), Standard unbiased estimate: Then: y(t)e ;i!t t=k+ y(t)y (t ; k) k 0 t= Standard biased estimate: = X N; k=;(n;) ^r(k)e ;i!k = ^ c (!) ^r(k) = N t=k+ y(t)y (t ; k) k 0 For both estimators: Consequence: Both ^ p (!) and ^ c (!) can be analyzed simultaneously. ^r(k) = ^r (;k) k < 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L 5

9 ; jkj N! ;i!k r(k)e Statistical Performance of ^ p (!) and ^ c (!) Bias Analysis of the Periodogram Summary: Both are asymptotically (for large N) unbiased: X N; E n^ p (!) o E o n^ (!) = c = k=;(n;) E f^r(k)g e ;i!k X N; E n^ p (!)o! (!) as N! = k=;(n;) = Both have large variance, even for large N. X k=; w B (k)r(k)e ;i!k Thus, ^ p (!) and ^ c (!) have poor performance. 8 ; jkj N jkj N ; < : w B (k) = Intuitive explanation: jkj N = Bartlett, or triangular, window 0 Thus, ^r(k) ; r(k) may be large for large jkj Even if f^r(k) ; r(k)g the errors are small, N; jkj=0 there are so many that when summed in ; p (!)], the PSD error is (!) large. [^ E n^ p (!) o = Z ; ()W B(! ; ) d Ideally: W B (!) = Dirac impulse (!). Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

10 Bartlett Window W B (!) Smearing and Leakage Main Lobe Width: smearing or smoothing 0 B = " sin(!n=) W (!) sin(!=) N B B (0), N = 5 (!)=W for W # Details in (!) separated in f by less than =N are not resolvable. φ(ω) smearing ^φ(ω) 0 0 </Ν ω ω db 0 Thus: Periodogram resolution limit = =N ANGULAR FREQUENCY Main lobe db width =N. For small N, W B (!) may differ quite a bit from (!). Sidelobe Level: leakage φ(ω) δ(ω) leakage ω ^ φ(ω) W B(ω) ω Lecture notes to accompany Introduction to Spectral Analysis Slide L 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L 9

11 Periodogram Bias Properties Periodogram Variance As N! E nh^ p (! ) ; (! )ih^ p (! ) ; (! ) io ( (! )! =! = =!! 0 Summary of Periodogram Bias Properties: Inconsistent estimate For small N, severe bias Erratic behavior As N!, W B (!)! (!), ^(!) so is asymptotically unbiased. ^φ(ω) ^ φ(ω) asymptotic mean = φ(ω) ω + st. dev = φ(ω) too Resolvability properties depend on both bias and variance. Lecture notes to accompany Introduction to Spectral Analysis Slide L 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L

12 NX t= y(t)w tk k = 0 ::: N ; ~NX NX f[y(t) ; y(t + ~N)]W t g ~W tp Discrete Fourier Transform (DFT) Radix Fast Fourier Transform (FFT) Assume: N = m X N= t= y(t)w tk + t=n=+ y(t)w tk Y (k) = Finite DTFT: Y N (!) = NX t= y(t)e ;i!t = X N= Nk + y(t + N=)W ]W tk [y(t) t= for even k! = Let N k and W = e;i N. W with Nk + = ; for odd k Y Then ( N k) N is the Discrete Fourier Transform (DFT): Let ~N = N= and ~W = W = e ;i= ~N. For k = 0 4 ::: N ; 4 = p: Direct fy (k)g computation of from : fy(t)gn t= N; k=0 flops ) O(N Y (p) = ~NX t= [y(t) + y(t + ~N)] ~W tp For k = 5 ::: N ; = p + : Y (k) = Each is a ~N = N=point DFT computation. t= Y (p + ) = Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

13 c k = k + c k; ) c k = kk FFT Computation Count Zero Padding Append the given data by zeros prior to computing DFT (or FFT): Let c k = number of flops for N = k point FFT. Goals: {z } N ::: y(n) 0 ::: 0g fy() Then Apply a radix FFT (so N = power of ) Finer sampling of ^(!): Thus, k N; N k N; N! c k = N log N ^φ(ω) k=0 continuous curve k=0 sampled, N=8 ω Lecture notes to accompany Introduction to Spectral Analysis Slide L 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L 5

14 BlackmanTukey Method Basic Idea: Weighted correlogram, with small weight applied to covariances ^r(k) with large jkj. X M; Improved PeriodogramBased Methods ^ BT (!) = k=;(m;) fw(k)g = Lag Window w(k)^r(k)e ;i!k Lecture w(k) M M k Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

15 BT (!) = ^ Z ^ p ()W (! ; )d BT (!) = ^ Z k=;(m;) w(k) Z BlackmanTukey Method, con't Window Design Considerations Nonnegativeness: p () {z } ^ W (! ; )d ; 0 ; W (!) = DTFTfw(k)g = Spectral Window Conclusion: ^ BT (!) = locally smoothed periodogram Effect: W (!) 0 (, w(k) If is a psd sequence) ^ Then: (!) 0 BT (which is desirable) TimeBandwidth Product Variance decreases substantially X M; Bias increases slightly N e = w(0) = equiv time width By proper choice of M: MSE = var + bias! 0 as N! ; W (!)d! W (0) = equiv bandwidth e = N e e = Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L 4

16 Window Design, con't Window Examples e = =N e = 0(=M) is the BT resolution threshold. As M increases, bias decreases and variance increases. Choose M as a tradeoff between variance and ) bias. Once M is given, N e (and hence e ) is essentially fixed. Choose window shape to compromise between ) smearing (main lobe width) and leakage (sidelobe level). The energy in the main lobe and in the sidelobes cannot be reduced simultaneously, once M is given. db Triangular Window, M = ANGULAR FREQUENCY db Rectangular M = 5 Window, ANGULAR FREQUENCY Lecture notes to accompany Introduction to Spectral Analysis Slide L 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L

17 ^ j (!) = M LX ^ B (!) = L : L j (k)e ;i!k 9 > = ^r > e;i!k Basic Idea: Bartlett Method Μ Μ... Ν Comparison of Bartlett and BlackmanTukey Estimates 8 >< φ (ω) ^ φ (ω) ^... average φ (ω) ^L LX X M; j= X M; >: k=;(m;) 8 < LX 9 = φ (ω) ^B = ^r j (k) k=;(m;) j= Mathematically: j (t) = y((j ; )M + t) t = ::: M y the jth subsequence = (j = ::: L 4 = [N=M]) Thus: ' X M; k=;(m;) ^r(k)e ;i!k B (!) ' ^ BT (!) with a rectangular ^ lag w window (k) R MX y j (t)e ;i!t Since ^ B (!) implicitly uses fw R (k)g, the Bartlett method has t= High resolution (little smearing) ^ B (!) = L ^ j (!) Large leakage and relatively large variance j= Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L 8

18 D (! k ) = ^ + J Welch Method Daniell Method Similar to Bartlett method, but allow overlap of subsequences (gives more subsequences, and thus better averaging) use data window for each periodogram; gives mainlobesidelobe tradeoff capability N subseq # subseq #... Let S =# of subsequences of length M. (Overlapping means S > [N=M] ) better averaging.) Additional flexibility: subseq #S By a previous result, for N, f^ p (! j )g are (nearly) uncorrelated random variables for j =! j N; N j=0 Idea: Local averaging of (J + ) samples in the frequency domain should reduce the variance by about (J + ). X k+j ^ p (! j ) The data in each subsequence are weighted by a temporal window j=k;j Welch is approximately equal to ^ BT (!) with a nonrectangular lag window. Lecture notes to accompany Introduction to Spectral Analysis Slide L 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L 0

19 D (!) ' ^ Daniell Method, con't Summary of Periodogram Methods Unwindowed periodogram reasonable bias unacceptable variance As J increases: Bias increases (more smoothing) Variance decreases (more averaging) Let = J=N. Then, for N, Z ; ^ p (!)d! Hence: ^ D (!) ' ^ BT (!) with a rectangular spectral window. Modified periodograms Attempt to reduce the variance at the expense of (slightly) increasing the bias. BT periodogram Local smoothing/averaging ^ of (!) p by a suitably selected spectral window. Implemented by truncating and weighting ^r(k) using a lag window in ^ c (!) Bartlett, Welch periodograms Approximate ^ interpretation: (!) BT with a suitable lag window (rectangular for Bartlett; more general for Welch). Implemented by averaging subsample periodograms. Daniell Periodogram Approximate ^ interpretation: (!) BT with a rectangular spectral window. Implemented by local averaging of periodogram values. Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

20 Basic Idea of Parametric Spectral Estimation Observed Data Assumed functional form of φ(ω,θ) Estimate parameters in φ(ω,θ) ^θ Estimate PSD ^φ(ω) =φ(ω,θ) ^ possibly revise assumption on φ(ω) Parametric Methods for Rational Spectra Lecture 4 Rational Spectra (!) = Pjkjm ke ;i!k Pjkjn ke ;i!k (!) is a rational function in e ;i!. By Weierstrass theorem, (!) can approximate arbitrarily well any continuous PSD, provided m and n are chosen sufficiently large. Note, however: choice of m and n is not simple Lecture notes to accompany Introduction to Spectral Analysis Slide L4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 some PSDs are not continuous

21 B(!) A(z) = + a z ; + + a n z ;n e (!) = nx nx mx i= a i r(k ; i) = mx j=0 b j E fe(t ; j)y (t ; k)g = m X j=0 b j h j;k AR, MA, and ARMA Models ARMA Covariance Structure By Spectral Factorization theorem, a rational (!) can be factored as ARMA signal model: (!) = A(!) i= a i y(t;i) = j=0 b j e(t;j) (b 0 = ) y(t)+ = + b z ; + + b m z ;m B(z) and, A(!) = A(z)j e.g., i! z=e Multiply by y (t ; k) and take E fg to give: Signal Modeling Interpretation: r(k) + e(t) y(t) B(q) B(!) y (!) = A(!) A(q) ltered white noise white noise ARMA: AR: MA: A(q)y(t) = B(q)e(t) A(q)y(t) = e(t) y(t) = B(q)e(t) H(q) = where B(q) = A(q) = 0 for k > m X k=0 h k q ;k (h 0 = ) Lecture notes to accompany Introduction to Spectral Analysis Slide L4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 4

22 " R " AR Signals: YuleWalker Equations AR Spectral Estimation: YW Method AR: m = 0. Writing covariance equation in matrix form for k = ::: n: YuleWalker Method: Replace r(k) by ^r(k) and solve for f^a i g and ^ : 4 r(;) ::: r(;n) r(0) r(0). r(). r(;)... ::: r(0) r(n) 4 5 a. a n 5 = ^r(;) ::: ^r(;n) ^r(0) ^r(0). ^r(). ^r(;)... ::: ^r(0) ^r(n) Then the PSD estimate is 4 5 ^a. ^a n 5 = 4 ^ # # These are the Yule Walker (YW) Equations. = ^(!) = j ^A(!)j ^ 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L4

23 +) y(n +) y(n i= a i y(t ; i) = y(t) + ' T (t) NX y(n ; ) y() y(n) +) y(n) y() y(n n 0 AR Spectral Estimation: LS Method Levinson Durbin Algorithm Fast, orderrecursive solution to YW equations Least Squares Method: nx 0 ; ;n ; n 5 = n 0. e(t) = y(t) y(t) + ^y(t) = where '(t) = [y(t ; ) ::: y(t ; n)] T. Find = [a ::: a n ] T to minimize {z } n+ R k = either r(k) or ^r(k). Direct Solution: t=n+ je(t)j f() = For one given value of n: O(n ) flops This gives ^ = ;(Y Y ) ; (Y y) where For k = ::: n: O(n 4 ) flops y = 4. y(n) 5 Y = 4. y(n ; ) y(n ; ) y(n ; n). 5 Levinson Durbin Algorithm: Exploits the Toeplitz form of R n+ to obtain the solutions for k = ::: nin O(n ) flops! Lecture notes to accompany Introduction to Spectral Analysis Slide L4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 8

24 Rx = y $ R~x = ~y, where ~x = [x n :::x ]T n ~r n+ ~r n 0 ~rn = 5 4 n 0 0 n 0 n + k n " ~ n 0 n + k n n ) n flops to determine f k kg n k= LevinsonDurbin Alg, con't LevinsonDurbin Alg, con't Relevant Properties of R: 4 n R n+ 0 4 n = 5 Combining these gives: 5 Rn+ 4 0 ~ n 4 n 0 5 = n 5 Nested structure R n+ 8 < : n ~ n 5 kn + 9 = 4 n + k n n 4 n+ = = 5 n+ n+ R 4 ṇ 4. 5 Thus, k n = ; n = n ) R n+ = # Thus, " n 0 # n+ = 4 n R n+ 0 n+ n+ R 4 = 5 4 n = 5 n+ = n + k n n = n ( ; jk nj ) 5 n ~r n+ ~r n 0 Computation count: 5 where k flops for the step k! k + n = n+ + ~r n n This is O(n ) times faster than the direct solution. Lecture notes to accompany Introduction to Spectral Analysis Slide L4 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 0

25 (!) = jb(!)j MA Signals MA Spectrum Estimation Two main ways to Estimate (!):. Estimate fb k g and and insert them in MA: n = 0 nonlinear estimation problem y(t) = B(q)e(t) ^(!) is guaranteed to be 0 = e(t) + b e(t ; ) + + b m e(t ; m) Thus,. Insert sample covariances f^r(k)g in: and (!) = mx k=;m r(k)e ;i!k r(k) = 0 for jkj > m (!) = jb(!)j = mx k=;m r(k)e ;i!k This is ^ BT (!) with a rectangular lag window of m + length. ^(!) is not guaranteed to be 0 Both methods are special cases of ARMA methods described below, with AR model order n = 0. Lecture notes to accompany Introduction to Spectral Analysis Slide L4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4

26 (!) = B(!) a j a p P mk=;m k e ;i!k ja(!)j P m k=;m ke ;i!k ja(!)j ARMA Signals ARMA Spectrum Estimation Two Methods: ARMA models can represent spectra with both peaks (AR part) and valleys (MA part). A(q)y(t) = B(q)e(t). fa Estimate b j g i (!) = in B(!) A(!) nonlinear estimation problem; can use an approximate linear twostage least squares method = where A(!) ^(!) is guaranteed to be 0. Estimate fa i r(k)g in (!) = k = E f[b(q)e(t)][b(q)e(t ; k)] g = E f[a(q)y(t)][a(q)y(t ; k)] g nx nx linear estimation problem (the Modified YuleWalker method). = r(k + p ; j) j=0 p=0 ^(!) is not guaranteed to be 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 4

27 = A(q) e(t) y(t) B(q) = ^ ; K N NX j^e(t)j = [a :::a n b :::b m ] T TwoStage LeastSquares Method TwoStage LeastSquares Method, con't Assumption: The ARMA model is invertible: = y(t) + y(t ; ) + y(t ; ) + Step : Replace fe(t)g by ^e(t) in the ARMA equation, = AR() with j k j! 0 as k! Step : Approximate, for some large K ' y(t) + y(t ; ) + + K y(t ; K) e(t) a) Estimate the f coefficients g K k= k by using AR modelling techniques. A(q)y(t) ' B(q)^e(t) and obtain estimates of fa i b j g by applying least squares techniques. Note that the a i and b j coefficients enter linearly in the above equation: b) Estimate the noise sequence y(t) ; ^e(t) ' [;y(t ; ) :::; y(t ; n) ^e(t ; ) :::^e(t ; m)] = y(t) + ^ y(t ; ) + + ^ K y(t ; K) ^e(t) and its variance t=k+ Lecture notes to accompany Introduction to Spectral Analysis Slide L4 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L4

28 Modified YuleWalker Method ARMA Covariance Equation: nx In matrix form for k = m + ::: m+ M..... " a 5. Replace fr(k)g by f^r(k)g and solve for faig. an. M = If n, fast Levinsontype algorithms exist for obtaining f^aig. M > n If Y W overdetermined system of equations; least squares solution for f^aig. Note: For narrowband ARMA signals, the accuracy of Lecture notes to accompany Introduction to Spectral Analysis Slide L4 r(k) + i= r(m) ::: r(m ; n +) r(m +) r(m ; n +) 4 r(m + M ; ) ::: r(m ; n + M) f^aig is often better for M > n air(k ; i) = 0 k > m # = ; r(m +) r(m +) 4 r(m + M) 5 Summary of Parametric Methods for Rational Spectra Computational Method Burden ^(!) 0 Accuracy? Use for AR: YW or LS Spectra with (narrow) peaks but no valley MA: BT Broadband spectra possibly with valleys but no peaks ARMA: MYW Spectra with both peaks and (not too deep) valleys ARMA: Stage LS mediumhigh mediumhigh Yes As above Guarantee low medium Yes low lowmedium No lowmedium medium No Lecture notes to accompany Introduction to Spectral Analysis Slide L4 8

29 ( ) ( ) ( ) Line Spectra Many applications have signals with (near) sinusoidal components. Examples: communications Parametric Methods for Line Spectra Part geophysical seismology ARMA model is a poor approximation Better approximation by Discrete/Line Spectrum Models radar, sonar Lecture 5 (!)!!!! ; An Ideal line spectrum Lecture notes to accompany Introduction to Spectral Analysis Slide L5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5

30 nx Z ; ei' d' nx Line Spectral Signal Model Covariance Function and PSD Signal Model: Sinusoidal components of frequencies k g and powers f kg, superimposed in white noise of f! power. Note that: E e i' p e ;i' o j =, for p = j n y(t) = x(t) + e(t) t = ::: E e pe ;i' o j = E n e i' o n p n ;i' o i' j e E k e i(! kt+ k ) {z } k= x(t) = Assumptions: = = 0, for p = j x k (t) A: k > 0! k [; ] (prevents model ambiguities) A: f' k g = independent rv's, uniformly distributed on [; ] (realistic and mathematically convenient) Hence, E n x p (t)x j (t ; k) o = p ei! pk p j r(k) = E fy(t)y (t ; k)g A: e(t) = circular white noise with variance and = P n p= pe i! pk + k 0 fe(t)e (s)g = t s E fe(t)e(s)g = 0 E (can be achieved by slow sampling) (!) = p= p(! ;! p ) + Lecture notes to accompany Introduction to Spectral Analysis Slide L5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 4

31 f! k k ' k g n k= NX nx Y = [y() :::y(n)] T k e i(! kt+' k ) Parameter Estimation Nonlinear Least Squares (NLS) Method Estimate either: (Signal Model) f! k k gn k= (PSD Model) Major Estimation Problem: f^! k g Once f^! k g are determined: f^ k g can be obtained by a least squares method from Let: t= y(t) ; k= {z } (! ') F k = k e i' k min k k ' k g f! ^r(k) = nx p= pe i^! pk + residuals = [ ::: n ] T OR: Both f^ k g and f^' k g can be derived by a least squares method from B = 4 i! e i! n e.. e in! e in! n 5 y(t) = nx k= k e i^! kt + residuals with k = k e i' k. Lecture notes to accompany Introduction to Spectral Analysis Slide L5 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5

32 F = (Y ; B) (Y ; B) = ky ; Bk ^ = (B B) ; B Y!=^! N k Nonlinear Least Squares (NLS) Method, con't NLS Properties Then: Excellent Accuracy: var (^! k ) = (for N ) [ ; (B B) ; B Y ] Example: N = 00 SNR k = k = = 0 db = [ ; (B B) ; B Y ] [B B] +Y Y ; Y B(B B) ; B Y This gives: Then q ) 0 k. ;5 var(^! and Difficult Implementation: = arg max ^!! Y B(B B) ; B Y The NLS cost function F is multimodal; it is difficult to avoid convergence to local minima. Lecture notes to accompany Introduction to Spectral Analysis Slide L5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 8

33 Y B(B B) ; B Y = N jy (!)j Unwindowed Periodogram as an Approximate NLS Method Unwindowed Periodogram as an Approximate NLS Method, con't For a single (complex) sinusoid, the maximum of the unwindowed periodogram is the NLS frequency estimate: Assume: n > Assume: n = Then: Then: B B = N B Y = NX (finite DTFT) f^! k g n k= the locations of the n largest ' peaks ^ of (!) p y(t)e ;i!t = Y (!) provided that t= ^ p (!) = (Unwindowed Periodogram) = So, with no approximation, ^ p (!) inf j! k ;! p j > =N which is the periodogram resolution limit. If better resolution desired then use a High/Super Resolution method. = arg max ^!! Lecture notes to accompany Introduction to Spectral Analysis Slide L5 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 0

34 LX nx HighOrder YuleWalker Method HighOrder YuleWalker Method, con't Recall: +e(t) y(t) = x(t) + e(t) = k= k e i(! kt+' k ) {z } k (t) x Degenerate ARMA equation for y(t): Estimation Procedure: ( ; e i! kq ; )x k (t) Let Estimate f^b i g L i= using an ARMA MYW technique = k n e i(! k t+' k ) ; e i! k e i[! k(t;)+' k ] o = 0 Roots of ^B(q) give f^! k g n k=, along with L ; n spurious roots. B(q) = + k= b k q ;k 4 = A(q) A(q) = ( ; e i! q ; ) ( ; e i! n q ; ) A(q) = arbitrary A(q) Then B(q)x(t) 0 ) B(q)y(t) = B(q)e(t) Lecture notes to accompany Introduction to Spectral Analysis Slide L5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5

35 LX i= b i r(k ; i) = 0 = UV U = (M n) with U U = I n V = (n L) with V V = I n HighOrder and Overdetermined YW Equations HighOrder and Overdetermined YW Equations, con't Fact: rank() = n ARMA covariance: SVD of : r(k) + k > L In matrix form for k = L + ::: L+ M ::: r() r(l) + ) ::: r() r(l.. b = ; + ) r(l + ) r(l. Thus, = (n n), diagonal and nonsingular r(l + M ; ) ::: r(m) r(l + M) {z } 4 = {z } 4 = )b = ; (UV The MinimumNorm solution is This is a highorder (if L > n) and overdetermined (if M > L) system of YW equations. b = ; y = ;V ; U Important property: The additional (L ; n) spurious zeros of B(q) are located strictly inside the unit circle, if the MinimumNorm solution b is used. Lecture notes to accompany Introduction to Spectral Analysis Slide L5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 4

36 HOYW Equations, Practical Solution Let ^ = but made from f^r(k)g instead of fr(k)g. Let ^U, ^, ^V be defined similarly to U,, V from the SVD of ^. Compute ^b = ;^V ^ ; ^U ^ Then f^! k g n k= are found from the n zeroes of ^B(q) that are closest to the unit circle. When the SNR is low, this approach may give spurious frequency estimates when L > n; this is the price paid for increased accuracy when L > n. Parametric Methods for Line Spectra Part Lecture Lecture notes to accompany Introduction to Spectral Analysis Slide L5 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L

37 a(!) = [ e ;i! ::: e ;i(m;)! ] T 0 n m The Covariance Matrix Equation Eigendecomposition of R and Its Properties Let: Note: rank(a) = n (for m n ) Let: R = AP A + I (m > n) A = [a(! ) ::: a(! n )] (m n) Define ::: m : eigenvalues of R ~y(t) 4 = 4 y(t) ; ) y(t. = A~x(t) + ~e(t) where y(t ; m + ) 5 :::s n g: orthonormal eigenvectors associated fs f with ::: n g ::: g m;n g: orthonormal eigenvectors associated fg f with ::: m g n+ = [x (t) :::x n (t)] T ~x(t) = [e(t) :::e(t ; m + )] T ~e(t) Then S = [s :::s n ] (m n) G = [g :::g m;n ] (m (m ; n)) with R 4 = E f~y(t)~y (t)g = AP A + I Thus, " S 5 G # R = [S G] P = E f~x(t)~x (t)g = 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

38 k > NX Eigendecomposition of R and Its Properties, con't MUSIC Method As rank(ap A ) = n: k = ::: n k = k = n + ::: m a (! ). 4 5 G = 0 A G = ; = nonsingular a (! n ) = 0 n ; Thus, ) fa(! k )g n k=? R(G) Note: f! k g n k= are the unique solutions of a (!)GG a(!) = 0. RS = AP A S + S = S 0 n Let: 5 S = A(PA S ; ) 4 = AC with jcj = 0 (since rank(s) = rank(a) = n). Therefore, since S G = 0, = ^R t=m N eigenvectors of ^R ^S ^G = S Gmade from the ~y(t)~y (t) A G = 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L 5

39 f^! k g n k= a(z) = [ z ; ::: z ;(m;) ] T Spectral and Root MUSIC Methods Pisarenko Method Spectral MUSIC Method: f^! k g n k= the locations of the n highest peaks of the = pseudospectrum function: Pisarenko is a special case of MUSIC with m = n + (the minimum possible value). If: m = n + Root MUSIC Method: a (!) ^G ^G a(!)! [; ] ^G = ^g Then:, f^! k g n k= can be found from the roots of ) a T (z ; )^g = 0 = the angular positions of the n roots of: no problem with spurious frequency estimates T (z ; ) ^G ^G a(z) = 0 a that are closest to the unit circle. Here, computationally simple Note: Both variants of MUSIC may produce spurious frequency estimates. (much) less accurate than MUSIC with m n + Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

40 a (!) " ^g T ; " a ) (z ^g MinNorm Method MinNorm Method, con't Spectral MinNorm Goals: Reduce computational burden, and reduce risk of false frequency estimates. f^!g n k= the locations of the n highest peaks in the = pseudospectrum # Uses m n (as in MUSIC), but only one vector in R(G) (as in Pisarenko). Root MinNorm = Let " ^g # = the vector in R( ^G), with first element equal to one, that has minimum Euclidean norm. f^!g n k= the angular positions of the n roots of the = polynomial # that are closest to the unit circle. Lecture notes to accompany Introduction to Spectral Analysis Slide L 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L 9

41 " # g m ; g R( ^G) ) ^S " ^g = A C = A DC = S C ; DC {z } S = (S S ); S S MinNorm Method: Determining ^g ESPRIT Method Let ^S = Then: Let A = [I m; 0]A S " ^g = 0 Then A = A D, where # # A = [0 I m; ]A e;i! ) S ^g = ; 5 MinNorm solution: ^g = ;S( S S) ; Also, let S = [I m; 0]S 0 e ;i! n D = As: I = ^S ^S = + S S, ( S S) ; exists iff S = [0 I m; ]S Recall S = AC with jcj = 0. Then = kk = (This holds, at least, N for.) Multiplying the above equation by gives: So has the same eigenvalues as D. is uniquely determined as ( ; kk ) = ( S S) ) ( S S) ; = =( ; kk ) ) ^g = ; S=( ; kk ) Lecture notes to accompany Introduction to Spectral Analysis Slide L 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L

42 ESPRIT Implementation From the eigendecomposition of ^R, find ^S, then ^S and ^S. The frequency estimates are found by: are the eigenvalues of ^ = ( ^S ^S ) ; ^S ^S ESPRIT Advantages: or Min Norm) no extraneous frequency estimates (unlike in MUSIC Lecture notes to accompany Introduction to Spectral Analysis Slide L where f^ k g n k= computationally simple f^! k g n k= = ; arg(^ k) accurate frequency estimates Summary of Frequency Estimation Methods Computational Accuracy / Risk for False Method Burden Resolution Freq Estimates Periodogram small mediumhigh medium Nonlinear LS very high very high very high YuleWalker medium high medium Pisarenko small low none MUSIC high high medium MinNorm medium high small ESPRIT medium very high none Recommendation: Use Periodogram for mediumresolution applications Use ESPRIT for highresolution applications Lecture notes to accompany Introduction to Spectral Analysis Slide L

43 Basic Ideas Two main PSD estimation approaches:. Parametric Approach: Parameterize (!) by a finitedimensional model. Filter Bank Methods. Nonparametric Approach: Implicitly smooth!=; by assuming that (!) is nearly f(!)g constant over the bands [! ;! + ] Lecture is more general than, but requires N > to ensure that the number of estimated values (= = = =)is< N. N > leads to the variability / resolution compromise associated with all nonparametric methods. Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L

44 Filter Bank Approach to Spectral Estimation! c! Division by with varying! c and xed bandwidth Filtered Calculation the band ;! QQ ;! c (a) ' ^FB(!) ; (a) consistent power calculation ()d= (b) ' jh()j (b) Ideal passband filter with bandwidth (c) () constant on [! ;! +] Note that assumptions (a) and (b), as well as (b) and (c), are conflicting. Lecture notes to accompany Introduction to Spectral Analysis Slide L y(t) y (!) Bandpass Filter! H(!) Z Signal ~y Power Power in Z!+!; lter bandwidth (c) ' (!) ()d= ^(! c ) Filter Bank Interpretation of the Periodogram 4 = ^p(~!) N = N where NX ;i~!t y(t)e NX i~!(n;t) y(t)e X y(n ; k) k h ( e i~!k k = 0 ::: N ; N otherwise 0 X h k e ;i!k = N Lecture notes to accompany Introduction to Spectral Analysis Slide L 4 h k = H(!) = k=0 = N t= t= k=0 center frequency of H(!) = ~! db bandwidth of H(!) ' =N e in(~!;!) ; e i(~!;!) ;

45 Filter Bank Interpretation of the Periodogram, con't Possible Improvements to the Filter Bank Approach 0 jh(!)j as a function of (~! ;!), for N = Split the available sample, and bandpass filter each subsample. more data points for the power calculation stage. db 5 0 This approach leads to Bartlett and Welch methods ANGULAR FREQUENCY. Use several bandpass filters on the whole sample. Each filter covers a small band centered on ~!. provides several samples for power calculation. Conclusion: The periodogram ^ p (!) is a filter bank PSD estimator with bandpass filter as given above, and: This multiwindow approach is similar to the Daniell method. narrow filter passband, power calculation from only sample of filter output. Both approaches compromise bias for variance, and in fact are quite related to each other: splitting the data sample can be interpreted as a special form of windowing or filtering. Lecture notes to accompany Introduction to Spectral Analysis Slide L 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L

46 mx mx ^(!) = = ^R ; m N NX Capon Method Capon Method, con't Capon Filter Design Problem: min(h Rh) subject to h a(!) = h Idea: Datadependent bandpass filter design. Solution: h 0 = R ; a=a R ; a The power at the filter output is: y F (t) = k=0 h k y(t ; k) [h 0 h ::: h m ] {z } = h 4 y(t). y(t ; m) 5 E n jy F (t)j o = h 0 Rh 0 = =a (!)R ; a(!) which should be the power of y(t) in a passband centered on!. {z } ~y(t) The ' Bandwidth m+ = (filter length) R = E f~y(t)~y (t)g E n jy F (t)j o = h Rh Conclusion Estimate PSD as: k=0 h k e ;i!k = h a(!) H(!) = where a(!) = [ e ;i! ::: e ;im! ] T with m + a (!) ^R ; a(!) ~y(t)~y (t) Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L 8 t=m+

47 + ; jkj m + m a ^Ra(!) (!) = + m BT (!) = a (!) ^Ra(!) ^ + m ^ C (!) = Capon Properties Relation between Capon and BlackmanTukey Methods Consider ^ BT (!) with Bartlett window: m is the user parameter that controls the compromise between bias and variance: ^ BT (!) = mx k=;m ^r(k)e ;i!k as m increases, bias decreases and variance increases. = + m mx t=0 mx s=0 ^r(t ; s)e ;i!(t;s) Capon uses one bandpass filter only, but it splits the Ndata point sample (N ; into m) subsequences of m length with maximum overlap. Then we have ^R = [^r(i ; j)] m + a (!) ^R ; a(!) Lecture notes to accompany Introduction to Spectral Analysis Slide L 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L 0

48 Relation between Capon and AR Methods Let AR k (!) = ^ k ^ j ^A k (!)j be the kth order AR PSD estimate of y(t). Then ^ C (!) = mx =^ AR Spatial Methods Part k (!) k=0 m + Consequences: Lecture 8 Due to the average over k, ^ C (!) generally has less statistical variability than the AR PSD estimator. Due to the loworder AR terms in the average, C (!) generally has worse resolution and bias ^ properties than the AR method. Lecture notes to accompany Introduction to Spectral Analysis Slide L Lecture notes to accompany Introduction to Spectral Analysis Slide L8

49 ; ; c c The Spatial Spectral Estimation Problem Simplifying Assumptions Source v B BB B BN cc c cc cc cc n v Source Farfield sources in the same plane as the array of sensors ; ; Nondispersive wave propagation ; ; Sensor Sensor m Problem: Detect and locate n radiating sources by using an array of m passive sensors. Emitted energy: Acoustic, electromagnetic, mechanical Receiving sensors: Hydrophones, antennas, seismometers Applications: Radar, sonar, communications, seismology, underwater surveillance Basic Approach: Determine energy distribution over space (thus the name spatial spectral analysis ) Hence: The waves are planar and the only location parameter is direction of arrival (DOA) (or angle of arrival, AOA). The number of sources n is known. (We do not treat the detection problem) The sensors are linear dynamic elements with known transfer characteristics and known locations (That is, the array is calibrated.) Lecture notes to accompany Introduction to Spectral Analysis Slide L8 Lecture notes to accompany Introduction to Spectral Analysis Slide L8

50 k (t) x(t ; k ) h jh k (! c )j(t)cos[! c t + '(t) ;! c k +argfh k (! c )g] ' k (t) = jh k (! c )j(t) y cos[! c t + '(t) ;! c k + arg H k (! c )] + e k (t) x(t) = k = Array Model Single Emitter Case the signal waveform as measured at a reference point (e.g., at the first sensor) the delay between the reference point and the kth sensor Narrowband Assumption Assume: The emitted signals are narrowband with known carrier frequency! c. Then: h k (t) = the impulse response (weighting function) of sensor k x(t) = (t)cos[! c t + '(t)] where (t) '(t) vary slowly enough so that e k (t) = noise at the kth sensor (e.g., thermal noise in sensor electronics; background noise, etc.) ; k ) ' (t) '(t ; k ) ' '(t) (t Time delay is ' now to a phase! shift k c : x(t ; k ) ' (t)cos[! c t + '(t) ;! c k ] Note: t R (continuoustime signals). Then the output of sensor k is k (t) = h k (t) x(t ; k ) + e k (t) y = convolution operator). ( Basic Problem: Estimate the time delays f k g with h k (t) known but x(t) unknown. This is a timedelay estimation problem in the unknown input case. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 4 where H k (!) = Ffh k (t)g is the kth sensor's transfer function Hence, the kth sensor output is Lecture notes to accompany Introduction to Spectral Analysis Slide L8 5

51 (t) = Complex Signal Representation Vector Representation for a Narrowband Source The noisefree output has the form: Let z(t) = (t) cos [! c t + (t)] = n e i[! c t+ (t)] + e ;i[! c t+ (t)] o Demodulate z(t) (translate to baseband): = the emitter DOA m = the number of sensors ;! ct = (t)f e i (t) {z } z(t)e lowpass e ;i[! ct+ (t)] {z } + g highpass a() = H (! c ) e ;i! c. 4 H m (! c ) e ;i! c m 5 Lowpass filter z(t)e ;i! ct to obtain (t)e i (t) Hence, by lowpass filtering and sampling the signal ~y k (t)= = y k (t)e ;i! ct Then y(t) = y (t). 4 y m (t) 5 e(t) = e (t). 4 e m (t) 5 y k (t) cos(! c t) ; iy k (t)sin(! c t) = we get the complex representation: t (for Z) y(t) = a()s(t) + e(t) or k (t) = (t) e i'(t) {z } y s(t) k (! c )j e i arg[h k (! c)] {z } jh H k (! c ) e ;i! c k + e k (t) NOTE: enters a() via both f k g and fh k (! c )g. For omnidirectional sensors the fh k (! c )g do not depend on. k (t) = s(t)h k (! c ) e ;i! c k + e k (t) y s(t) where is the complex envelope of x(t). Lecture notes to accompany Introduction to Spectral Analysis Slide L8 Lecture notes to accompany Introduction to Spectral Analysis Slide L8

52 Analog Processing Block Diagram Analog processing for each receiving array element SENSOR DEMODULATION X XXXXX Xz : External x(t ; k) Transducer Filtering Internal Noise cos(!ct) sin(!ct) Oscillator Im[~yk(t)] A/D A/D Lecture notes to accompany Introduction to Spectral Analysis Slide L8 8 Noise Incoming Signal l ll,,,? Cabling and y k (t)? Re[~yk(t)] Lowpass Filter Lowpass Filter Re[yk(t)] Im[yk(t)] A/D CONVERSION!!!! Multiple Emitter Case Given n emitters with received signals: fs k (t)g n k= DOAs: k Linear sensors ) Let Then, the array equation is: Use the planar wave assumption to find the dependence of k on. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 9 y(t) = a( )s (t) + + a(n)sn(t) + e(t) A = [a( ) :::a(n)] (m n) s(t) = [s(t) :::sn(t)] T (n ) y(t) = As(t) + e(t)

53 J J^ JJ J JJ J JJ J] J sin J^d J JJ 4 d sin!!c = s c d sin = sin j dj d = = Uniform Linear Arrays Spatial Frequency h J JJ Let: Source d sin = c c=f = c=f c = signal wavelength + J JJ J JJ v v v v v? ] 4 p p p Sensor a() = [ e ;i! s ::: e ;i(m;)! s ] T By direct analogy with the vector a(!) made from uniform samples of a sinusoidal time series, m d ULA Geometry Sensor # = time delay reference! s = spatial frequency The function! s! a() is onetoone for Time Delay for sensor k: As j! s j $ d = spatial sampling period k = (k ; ) d sin c c where = wave propagation speed d = is a spatial Shannon sampling theorem. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L8

54 Spatial Filtering Spatial filtering useful for Spatial Methods Part DOA discrimination (similar to frequency discrimination of timeseries filtering) Lecture 9 Nonparametric DOA estimation There is a strong analogy between temporal filtering and spatial filtering. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9

55 h = [h o :::h m; ] y(t) = [u(t) :::u(t ; m + )] T mx Analogy between Temporal and Spatial Filtering Analogy between Temporal and Spatial Filtering Spatial Filter: Temporal FIR Filter: fy k (t)g m k= the spatial samples obtained with a = sensor array. X m; y F (t) = h k u(t ; k) = h y(t) Spatial FIR Filter output: k=0 If u(t) = e i!t then Narrowband Wavefront: The array's (noisefree) response to a narrowband ( sinusoidal) wavefront with complex envelope s(t) is: k= h k y k (t) = h y(t) y F (t) = {z } filter transfer function F (t) = [h a(!)] u(t) y = y(t) a()s(t) c ;i! c ;i! m e ] T ::: [ e = a() a(!) = [ e ;i! ::: e ;i(m;)! ] T We can select h to enhance or attenuate signals with different frequencies!. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 The corresponding filter output is {z } filter transfer function F (t) = [h a()] s(t) y We can select h to enhance or attenuate signals coming from different DOAs. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 4

56 u(t) =e i!t? s q ; q ; Z ZZ! a a!! a a! B e ;i! C {z } a(!) B. e ;i!m() {z } a() h 0 h h [h a(!)]u(t) [h a()]s(t) Analogy between Temporal and Spatial Filtering Spatial Filtering, con't? 0 z B? C C B C B C s? B m ; q? q B B e ;i(m;)! C C C A u(t) h m; Example: The response magnitude jh a()j of a spatial filter (or beamformer) for a 0element ULA. Here,? h = a( 0 ), where 0 = 5 0 (a) Temporal filter 0 0 Theta (deg) (Temporal sampling) source with DOA= narrowband z 0 0 Z~? reference point 0 C! a a! B B e ;i!() q? P q C C C A s(t) Magnitude m? (Spatial sampling) (b) Spatial filter Lecture notes to accompany Introduction to Spectral Analysis Slide L9 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L9

57 Spatial Filtering Uses Nonparametric Spatial Methods A Filter Bank Approach to DOA estimation. Basic Ideas Spatial Filters can be used To pass the signal of interest only, hence filtering out interferences located outside the filter's beam (but possibly having the same temporal characteristics as the signal). To locate an emitter in the field of view, by sweeping the filter through the DOA range of interest ( goniometer ). Design a filter h() such that for each It passes undistorted the signal with DOA = It attenuates all DOAs = Sweep the filter through the DOA range of interest, and evaluate the powers of the filtered signals: E n jy F (t)j o = E n jh ()y(t)j o h ()Rh() = R = E fy(t)y with (t)g. The (dominant) peaks of h ()Rh() give the DOAs of the sources. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 8

58 NX Beamforming Method Implementation of Beamforming Assume the array output is spatially white: Then: E n jy F (t)j o = h h = ^R N The beamforming DOA estimates are: t= y(t)y (t) R = E fy(t)y (t)g = I Hence: In direct analogy with the temporally white assumption for filter bank methods, y(t) can be considered as impinging on the array from all DOAs. Filter Design: Solution: min (h h) subject to h a() = h f^ k g = the locations of the n largest peaks of a () ^Ra(). This is the direct spatial analog of the BlackmanTukey periodogram. Resolution Threshold: inf j k ; p j > wavelength array length = array beamwidth h = a()=a ()a() = a()=m E n jy F (t)j o = a ()Ra()=m Inconsistency problem: Beamforming DOA estimates are consistent if n =, but inconsistent if n >. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 0

59 Capon Method Parametric Methods Filter design: Assumptions: The array is described by the equation: min(h Rh) subject to h a() = h Solution: The noise is spatially white and has the same power in all sensors: y(t) = As(t) + e(t) h = R ; a()=a ()R ; a() E n jy F (t)j o = =a ()R ; a() Implementation: The signal covariance matrix E fe(t)e (t)g = I is nonsingular. P = E fs(t)s (t)g f^ k g = the locations of the n largest peaks of =a () ^R ; a(): Performance: Slightly superior to Beamforming. Both Beamforming and Capon are nonparametric approaches. They do not make assumptions on the covariance properties of the data (and hence do not depend on them). Lecture notes to accompany Introduction to Spectral Analysis Slide L9 Then: R = E fy(t)y (t)g = AP A + I Thus: The NLS, YW, MUSIC, MINNORM and ESPRIT methods of frequency estimation can be used, almost without modification, for DOA estimation. Lecture notes to accompany Introduction to Spectral Analysis Slide L9

60 NX Nonlinear Least Squares Method Nonlinear Least Squares Method min k g fs(t)g f t= ky(t) ; As(t)k N {z } f( s) f Minimizing s over gives Properties of NLS: ^s(t) = (A A) ; A y(t) t = ::: N Then Performance: high f( ^s) = N NX t= k [I ; A(A A) ; A ]y(t)k Computational complexity: high = N NX t= y (t)[i ; A(A A) ; A ]y(t) Main drawback: need for multidimensional search. Thus, = trf[i ; A(A A) ; A ] ^Rg k g = arg max f^ k g trf[a(a A) ; A ] ^Rg f For N =, this is precisely the form of the NLS method of frequency estimation. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 4

61 " y(t) ~y(t) {z} U ] M;n ^; = N NX YuleWalker Method YWMUSIC DOA Estimator " A ~A # " e(t) + s(t) # # y(t) = = f^ k g = the n largest peaks of =~a ()^V ^V ~a() ~e(t) Assume: E fe(t)~e (t)g = 0 where Then: ~a(), (L ), is the array transfer vector for ~y(t) at DOA ; 4 = E fy(t)~y (t)g = AP ~A (M L) ^V is defined similarly to V, using Also assume: y(t)~y (t) t= Properties: M > n L > n ( ) m = M + L > n) rank( A) = rank( ~A) = n Computational complexity: medium Then: rank(;) = n, and the SVD of ; is Performance: satisfactory if m n Main advantages: " 0 nn 0 0 " # V V # n g L;n g = [ U {z} ; n weak assumption on fe(t)g the subarray A need not be calibrated Properties: ~A V = 0 V R( ~A) Lecture notes to accompany Introduction to Spectral Analysis Slide L9 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L9

62 MUSIC and MinNorm Methods ESPRIT Method Assumption: The array is made from two identical subarrays separated by a known displacement vector. Both MUSIC and MinNorm methods for frequency estimation apply with only minor modifications to the DOA estimation problem. Spectral forms of MUSIC and MinNorm can be used for arbitrary arrays Let m = # sensors in each subarray A = [I m 0]A (transfer matrix of subarray ) A = [0 I m ]A (transfer matrix of subarray ) Root forms can be used only with ULAs Then A = A D, where MUSIC and MinNorm break down if the source signals are coherent; that is, if rank(p ) = rank(e fs(t)s (t)g) < n Modifications that apply in the coherent case exist. D = e;i! c( ) e ;i! c( n ) = the time delay from subarray to () subarray for a signal with DOA = : 5 () = d sin()=c where d is the subarray separation and is measured from the perpendicular to the subarray displacement vector. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 8

63 ESPRIT Method, con't ESPRIT Scenario source θ subarray known displacement vector subarray Properties: Requires special array geometry Computationally efficient No risk of spurious DOA estimates Does not require array calibration Note: For a ULA, the two subarrays are often the first m ; and last m ; array elements, so m = m ; and A = [I m; 0]A A = [0 I m; ]A Lecture notes to accompany Introduction to Spectral Analysis Slide L9 9